Here case \(\mathbf {E}\)  is no longer real-valued. This may be justified by extending the ansatz to
\(\mathbf {E}(\mathbf {x},t)= \mathbf {U}(\mathbf {x})e^{\mathrm {i}\kappa t}+c.c.\)  and by either neglecting higher harmonics generated from the cubic nonlinearity or by assuming the time-averaged constitutive relation
\(\mathbf {P}(\mathbf {E})=\varepsilon _0\chi _1(\mathbf {x})\mathbf {E}+\varepsilon _0\chi _3(\mathbf {x})\frac{1}{T}\int _0^T{\mathbf {E}}^2\,dt \mathbf {E}\)  with \(T=2\pi /\kappa \) , cf. [1]}, [2]}. For results on (REF ) we refer to [3]}, [4]} and in particular to the survey [5]}. Time-harmonic traveling waves have been found in a series of papers [6]}, [1]}, [8]}. The number of results for monochromatic standing polarized wave profiles \(U(\mathbf {x})=(0,u(x_1),0)\)  with \(u\)  satisfying \(0=-u^{\prime \prime }-\kappa ^2\left( V(x_1)u+\Gamma (x_1)|u|^2u\right)\)  on \(\mathbb {R}\)  is too large to cite so we restrict ourselves to Cazenave's book [9]}.
